MD11 FLC OPERATES ACFT IN CLASS II AIRSPACE WITH 1 FMC INOP.

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I WAS ASSIGNED AS FO ON THE SECOND LEG OF FLT FOLLOWING ACCEPTANCE OF THE ACFT FROM 'A' CHK. AS WE PARKED THE ACFT ATZ; THE ACFT REJECTED THE GND PWR AS IT WAS SWITCHED FROM APU TO GPU. THIS RESULTED IN THE ACFT DEPWRING WITH THE RESULTANT SWITCH TO EMER PWR; AND THE SYS SHUTTING DOWN. WE REAPPLIED THE APU AND THE ACFT SYS BEGAN TO REBOOT. THE #1 FMC; HOWEVER; WOULD NOT RESET. WE CALLED FOR OUR MAINT; RPTED THE MALFUNCTION; PUT IT IN THE ACFT MAINT LOG AND THEY BEGAN TO TROUBLESHOOT THE SYS. WE WERE TOLD THAT A PART WOULD BE SUPPLIED AT A DOWNLINE STATION. WE WERE SENT AN AMENDED RELEASE REFLECTING MEL. WE COMPLIED WITH THE MAINT AND OPERATIONAL PROCS TO PROCEED. WE DEPARTED AND COMPLETED THE SEGMENT UNEVENTFULLY. WE TURNED THE ACFT OVER TO MAINT AND THEY TRIED TO RESET THE INOP COMPUTER. THE NEXT CREW ALSO ACCEPTED THE ACFT AND DEPARTED. THE FLT WAS CONDUCTED UNDER PROCS THAT WE WERE TRAINED TO; AND DID; FOLLOW. WE COMPLIED WITH THE PROCS AS SPELLED OUT IN THE MEL; FCOM. I SHOULD HAVE REJECTED THE ACFT FOR CLASS 2 NAV.